Rational Numbers Test

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Rational Numbers Test - 10

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Question 1
1 / -0

What is the value of x in the two equivalent rational numbers and?

A
45

B
30

C
48

D
27

Solution

As the numbers are equivalent,
=

=

=

So, x = 45
Question 2
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What is the sum of the three middle rational numbers if are arranged in descending order?

A

B

C

D

Solution

To arrange in descending order, denominators should be the same.

The LCM of 3, 2, 6 and 4 is 12.

After arranging in descending order

Sum of the three middle terms =
Question 3
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The product of two rational numbers is . If one of the numbers is , then what is the other number?

A

B

C

D

Solution

Let the other rational number be 'x'.
Then,
Question 4
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Which of the following statements is true for the rational number ?

A
It lies to the left side of 0 on the number line.

B
It lies to the right side of 0 on the number line.

C
It is not possible to represent this number on a number line.

D
None of these

Solution

As is a negative number and we know that negative numbers lie to the left side of 0 on the number line; lies to the left side of 0 on the number line.
Question 5
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Which of the following is/are correct?

(1)

(2)

A
Only (1)

B
Only (2)

C
Both (1) and (2)

D
Neither (1) nor (2)

Solution

To compare any two rational numbers, we will make the denominators equal first.

So we can say that

Also,
We can say that
Both (1) and (2) are correct.
Question 6
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Which of the following statements is/are true?

A
Negative numbers are greater than positive numbers.

B
The additive inverse of the rational number is ^.

C
Rational numbers cannot be represented on a number line.

D
All of the above

Solution

The additive inverse of the rational number is .
This statement is true.

The other two statements are not true because 'negative numbers are smaller than positive numbers' and 'rational numbers can be represented on a number line'.
Question 7
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Consider the following statements (P) and (Q):
(P): The number 0 is neither a positive nor a negative rational number.
(Q): If the numerator and denominator of a rational number are multiplied or divided by a non-zero integer, we get a rational number which is said to be equivalent to the given rational number.

Which of the following options holds true?

A
(P) is true and (Q) is false.

B
(P) is false and (Q) is true.

C
Both (P) and (Q) are true.

D
Both (P) and (Q) are false.

Solution

(P): The number 0 is neither a positive nor a negative rational number.
The above statement (P) holds true because '0' is neither a positive nor a negative rational number.

(Q): If the numerator and denominator of a rational number are multiplied or divided by a non-zero integer, we get a rational number which is said to be equivalent to the given rational number.

The above statement (Q) holds true. We can prove it with the help of the following example:

Also,
So, is the equivalent form of .
Question 8
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If , then what is the value of ?

A

B

C

D

Solution

=
=
=
=
=
Question 9
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Fill in the blanks:

1) _______ numbers consist of all integers, fractions and zero.
2) There are _________ rational numbers between two rational numbers.
3) If a real number X added to –X is zero, then -X is known as the _______ inverse of X.

A
Whole; zero; additive

B
Rational; finite; subtractive

C
Rational; infinite; additive

D
Rational; infinite; multiplicative

Solution

1) Rational numbers consist of all integers, fractions and zero.
2) There are infinite rational numbers between two rational numbers.
3) If a real number X added to –X is zero, then -X is known as the additive inverse of X.
Question 10
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Which of the following subtractions is not in its simplest form?

A

B

C

D

Solution

(1)
=
is in its simplest form.
(2)
=
is in its simplest form.
(3)
=
is in its simplest form.
(4)
=
is not in its simplest form.
Question 11
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Simplify:

A

B

C

D

Solution
Question 12
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Which of the following number lines correctly represents the rational number ^?

A

B

C

D

Solution

=
So, will lie between -1 and 0.
Question 13
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In which of the following options are the numbers arranged in ascending order?

A
,,,

B
,,,

C
,,,

D
,,,

Solution

First we need to make the denominators same by taking the LCM of 7, 8, 9 and 9, which is equal to 504.

Now, = =
= =
= =
= =
< < <
Hence, the numbers in option 2 are in ascending order.
Question 14
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Which of the following options represents equivalent rational numbers?

A
and

B
and

C
and

D
and

Solution

We know that If is a rational number and m is a non-zero integer, then is a rational number equivalent to . and are equivalent rational numbers as:
=
Question 15
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What will be the result if we divide the sum of and by the difference of and ?

A

B

C

D

Solution

Sum of and :
- = =

Difference of and :
- = =

According to the question,
÷ = × =
Question 16
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A carton is to be filled with packs of spices, each weighing kg. The weight of the carton should not exceed kg. Find the largest number of packs of spices that can be put inside the carton.

A
2

B
4

C
6

D
10

Solution

Weight of each pack of spices = kg

Carrying capacity of the carton = kg

Largest number of packs of spices =
Question 17
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Rohan brought a cake as it was the birthday of his friend. Riya ate of it, Priya ate of it and the rest of the cake was eaten by Rohan. What part of the cake has Rohan eaten?

A

B

C

D

Solution

Part of the cake eaten by Riya =
Part of the cake eaten by Priya =
Part of the cake eaten by Rohan =
Question 18
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A seminar was held in an auditorium consisting of doctors, engineers, lawyers and teachers. There were a total of 1500 people in the auditorium. If of the total people were doctors, of the total people were engineers , of the total people were lawyers and rest of the total people were teachers., then how many teachers were there?

A
40

B
64

C
105

D
120

Solution

Number of doctors = × 1500 = 350

Number of engineers = × 1500 = 450

Number of lawyers = × 1500 = 660

Number of teachers = 1500 - (350 + 450 + 660)
= 1500 - 1460 = 40
Question 19
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Three shopkeepers P, Q and R buy kg of peanuts altogether. Then they divide it equally among them. How many kilograms of peanuts will shopkeeper P get?

A
kg

B
kg

C
kg

D
kg

Solution

Total quantity of peanuts = kg = kg
So, each shopkeeper will get:

÷ 3
= × = kg or kg

So, shopkeeper P will get kg.
Question 20
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Raman leaves for college from home and after covering km of distance, he realises that he has forgotten his wallet at home. So he goes back towards the home, but after covering km, his bike stops working due to some mechanical failure. How far away is he from his home now?

A
km

B
km

C
km

D
km

Solution

Distance covered by Raman before he realises that he has forgotten his wallet = km
Distance covered by Raman before his bike stops working = km
Distance between Raman's house and the location where his bike stops working = ( - ) = = km

Question 21

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Match the following:

(1) On dividing the sum of and by the product of and , we get ______.	(a)
(2) Sahib spends ^th of his income. He spends ^rd of it on books, ^th on grocery and the rest on clothes. He spends ____ part on clothes.	(b) 12
(3) 64 shirts can be made from _______ metres of cloth if each shirt requires metres of cloth.	(c) 16
(4) Three bundles of potatoes weigh kg, kg and kg. The total weight of potatoes is ___ kg.	(d)

(1) - (d), (2) - (a), (3) - (c), (4) - (b)

(1) - (d), (2) - (a), (3) - (b), (4) - (c)

(1) - (a), (2) - (d), (3) - (c), (4) - (b)

(1) - (d), (2) - (b), (3) - (c), (4) - (a)

Solution

(1) = =
(2) = = =
(3) Cloth required for 64 shirts = = 16 metres
(4) =

Question 22
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Which of the following statements is/are correct?

(1) The sum of the two rational numbers and is .

(2) is the smallest of the three rational numbers ^.
(3) While multiplying a rational number by a negative integer, we multiply the numerator by that integer, keeping the denominator unchanged.
(4) The product of a rational number and its reciprocal is always 1.

A
1, 3 and 4

B
All 1, 2, 3 and 4

C
Both 2 and 4

D
Only 1

Solution

(1) Statement 1 is correct.

(2) Statement 2 is correct.

So, is the smallest of the three fractions.

(3) Statement (3) is also correct.
For example,

While multiplying a rational number (-2/9) by a negative integer (-5), we multiply the numerator (-2) by that integer (-5), keeping the denominator (9) unchanged.

(4) Statement (4) is also correct.
Question 23
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Consider the following statements and choose the correct option:

1. and are the equivalent forms of the rational numbers and , respectively.

2. Every integer and every whole number is a rational number.

A
Both statement 1 and statement 2 are true.

B
Both statement 1 and statement 2 are false.

C
Statement 1 is true and statement 2 is false.

D
Statement 1 is false and statement 2 is true.

Solution

(1)

So, and are equivalent rational numbers.

And

So, and are equivalent rational numbers.

Therefore, statement 1 is correct.

Statement 2 is correct because every integer and every whole number is a rational number.
All integers and whole numbers can be represented in the form of p/q, where p and q are integers and q ≠ 0.
Question 24
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State "True" or "False" for the following statements.

1. '0' is a positive rational number.
2. A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.
3. All the rational numbers can be represented on a number line.

4. should be subtracted from so as to get

A
(1) - F, (2) - T, (3) - T, (4) - T

B
(1) - F, (2) - T, (3) - F, (4) - T

C
(1) - T, (2) - F, (3) - F, (4) - T

D
(1) - T, (2) - T, (3) - F, (4) - F

Solution

(1) The number '0' is neither a positive nor a negative rational number.
So, statement (1) is false.
(2) A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.
This statement is true.
For example, rational numbers like are in the standard form.

(3) All the rational numbers can be represented on a number line.
This statement is also true.

(4)
So, statement (4) is also true.
Question 25
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Fill in the blanks:

1. The division of a rational number by ____(P)____ is not defined.

2. ___(Q)___ should be added to so as to get^.3. If there are two rational numbers with a common denominator, then the rational number with the larger numerator is ____(R)____ than the other rational number.
4. To reduce the rational number to its standard form, we divide its numerator and denominator by their _____(S)____ ignoring the negative sign.

A
(P): 0, (Q): , (R): greater, (S): HCF

B
(P): 0, (Q): , (R): smaller, (S): LCM

C
(P): 0, (Q): , (R): smaller, (S): LCM

D
(P): -1, (Q): , (R): greater, (S): HCF

Solution

(1) Division by 0 is not defined.
For example, - = not defined

(2) Let x be the required number.
So,

(3) If there are two rational numbers with a common denominator, then the rational number with the larger numerator is greater than the other rational number.
For example,

(4) To reduce a rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign.